@Emily1986xx Great news about the hospital! It will still likely take a while for an appointment. In the meantime, do take pictures and date them so you can show what has happened and why you are so concerned.
With it being around the mouth, could be eczema irritated by something she is eating. At this age, things like tomatoes and certain other fruits can being the redness. It is not a 'true' allergic reaction in the same way and should help by cleaning around the mouth, if memory serves me right. The picture of her back though looks like uticaria or hives. Allergy UK have a lot of useful info https://www.allergyuk.org/ and Anaphylaxis.org have a lot of really good factsheets: https://www.anaphylaxis.org.uk/factsheets/ I would say it would be worthwhile reading and learning and arming yourself with some info. Most GPs barely cover any allergy or eczema in their training so it won't take long before you know as much as they do.
Also worth educating yourself on the treatments. Some of the steroids are not so great, especially in sunlight. Certainly there are some I would never use. Understand your options. If it is allergy-related, you need to look at antihistamines. The pharmacy can probably help with that and I would definitely give my child them today with the skin like that. But, not sure what can be given over the counter for that age these days, you might need to wait for a prescription. You can always get the pharmacist to tell you what they would recommend and then ask the GP directly for that until they can refer you or tell you why it is happening, etc.
I would say the usual suspects are milk, egg, nuts, peanuts, soy or sesame at this age. It doesn't have to be every time she has the food as it could be the format of it (e.g. milk baked in a biscuit is different to drinking milk). Eczema reactions are also a little slower, so could be something she had the half day or whole day before. The other thing to look at is contact with something - laundry detergent, cream, pet, etc. Keep a record of foods (I know, boring and a pain for sure) and also of her mood - is she irritated or sort of fussing or whiny or sleepy after eating something particular? Tummy aches/poos that are different? Does she itch the redness?
Don't eliminate anything unless you are sure it is the culprit. Just write down what she has and when and then what happens, and when.
It doesn't sound at all like the steroids were the problem. The GPs give much stricter instructions than the consultants usually do. The reality is, they are usually not tricky to use unless it is really long term (though, some particular ones might be, but I would guess the GP gave you something pretty broad as a starter). Usually with steroids it should be a kind of 'hard and fast' approach to combat a flare up, then taper off the usage. The only way a steroid would have caused the hives is if she is allergic to the ingredients. It is more likely that it calmed it down, but the cause is yet unknown so it has come back.
For the GP - be insistent. Do you have any atopic members of the family? This would help your case (anaphylaxis, eczema, asthma, hayfever, etc.) that it should be taken seriously. You could also talk to one of the organisations in the meantime which would also arm you with facts and info to help you but also to get seen. I think GPs are usually ok with referring you as it gets you off their case. They just have to have enough of a reason to justify it. I think the NICE guidelines used to be about saying the right thing. If, for example, you tell them it is affecting the daily lives of the child, you, the whole household, they are more likely to refer you.